I need urgent help in determining the final result of the definite integral:
$$\int_0^{2\pi}\left (b\cos(x) + \sqrt{a^2 - b^2\sin^2(x)}\right)\, \mathrm{d}x.$$ I know, by using an online integral calculator, the indefinite integral gives:
$$|a| E\left(x\left|\frac{b^2}{a^2}\right.\right) + b \sin(x) + C$$
But, since it is an elliptical integral (of the second kind with parameter $m=k^2$), which is out of the scope of the mathematics I have studied and been familiar with, I cannot literally do anything about it.
Could someone, acquainted with elliptic integrals, apply the boundaries of the integral for me, please?
Please, I need this so badly. Any help would be appreciated.
Note that $$\int_0^{2\pi}\sqrt{a^2-b^2\sin^2x}\,dx=4|a|\int_0^{\pi/2}\sqrt{1-(b/a)^2\sin^2x}\,dx=4|a|E(k=b/a)$$ and this is the complete elliptic integral. The other term in the original integral evaluates to zero, so the final answer is $$4|a|E(b/a)$$